## Inverse Operations: An Introduction

Inverse operations are operations that have opposing or contrary results. The result that we get can be verified by using the inverse operations. Operations like addition, subtraction, division and multiplication have their inverse operations. The inverse operation of subtraction is addition and the inverse operation of division is multiplication.

## What is the Inverse of Division?

The inverse property of division is multiplication. The division consists of a dividend, a divisor and a quotient. If you multiply the quotient with the divisor, you will get the dividend as the answer.

For example, \[15 \div 3 = 5\]. Here, if we multiply 5 and 3 by each other, we will get the result of 15. This shows that the inverse of division is multiplication.

## What is the Inverse of Multiplication?

The inverse of multiplication is division. If we multiply the two numbers, then the result can be used for the division as the inverse of multiplication.

For example, \[4 \times 5 = 20\] here if we divide 20 by either 4 or 5, the result we will get would be 5 or 4, respectively. This shows that the inverse of multiplication is division.

## What is the Inverse of Subtraction?

The inverse of subtraction is addition. The result when added to one of the numbers will result in the other number.

For example, \[30 - 20 = 10\] here if you add 20 to 10, then it will result in 30. So, this proves that addition is the inverse of subtraction.

## What is the Inverse of Addition?

The inverse of addition is subtraction. The result can be subtracted from the number and it will give us the other number.

For example, \[12 + 15 = 27\] if we subtract 27 from 15, then we will get an answer of 12. This shows that subtraction is the inverse of addition.

## Inverse Operation Characteristics

**Property of Inverse Addition:**The additive inverse is the value that, when added to the original integer, yields 0.**Property of Inverse Multiplication:**The multiplicative inverse is the value that, when multiplied by the original integer, yields 1.

## Example of Additive Inverse Property

If x is the original integer, then its additive inverse is negative x, i.e., -x. If -x is the starting point, then the additive inverse will be the positive value of x, i.e., x. For example, the additive inverse of -10 would be 10 and the additive inverse of 8 would be -8.

## Example of Multiplicative Inverse Property

If x is the original integer, then its multiplicative inverse is \[\dfrac{1}{x}\] and if the original integer is \[\dfrac{1}{x}\], then its multiplicative inverse will be x. If the original integer and its multiplicative are multiplied, then the result would also be 1.

For example, the multiplicative inverse of 5 would be \[\dfrac{1}{5}\] and if we multiply these two, then it would result in 1 as the answer.

## Sample Questions

1. The inverse operations of \[12 + 56 = 68\] would be

a. \[68 - 56\]

b. \[68 - 12\]

c. none of the above

d. A and B

Ans: A and B

Explanation: The inverse of addition is subtraction. So, the inverse of \[12 + 56 = 68\] would be \[68 - 56\] and \[68 - 12\].

2. The inverse of multiplication is

a. addition

b. subtraction

c. division

d. all of the above

Ans: Division

3. The inverse of operation is basically

a. opposite

b. same

c. exact

d. none of the above

Ans: Opposite

## Conclusion

The inverse operation will change from addition to subtraction and from division to multiplication and vice versa. The inverse of multiplication when multiplied with the result gives 1 as the answer. The inverse operations help in verifying the answers.

## FAQs on Inverse Operations

1. Write the inverse operation of 54 - 12 = 42.

The operation given is in the subtraction form \[54 - 12 = 42\].

The inverse of subtraction is addition, so the result is \[42 + 12 = 54\].

2. Write the multiplicative inverse for $5 + \dfrac{6}{{10}}$

The multiplicative inverse for \[5 + \dfrac{6}{{10}}\] would be

\[\begin{array}{l} = \dfrac{{50 + 6}}{{10}}\\ = \dfrac{{56}}{{10}}\end{array}\]

The multiplicative inverse is \[\dfrac{{10}}{{56}}\].

3. What is the additive inverse of -43?

The number given is in the negative form. So, the additive inverse will make it positive. The additive inverse of -43 would then be +43.

4. What will be the additive inverse of 5 x 2 ?

The answer for \[5 \times 2\] is 10. So, the additive inverse of 10 would be -10.

5. What would be the inverse for 3^{2}

The inverse of the square operation would be the square root of the number.

\[\begin{array}{l}{3^2} = 9\\\sqrt 9 = 3\end{array}\].